Question: Solve for $x$ : $ 4|x + 4| - 6 = -1|x + 4| + 7 $
Explanation: Add $ {1|x + 4|} $ to both sides: $ \begin{eqnarray} 4|x + 4| - 6 &=& -1|x + 4| + 7 \\ \\ { + 1|x + 4|} && { + 1|x + 4|} \\ \\ 5|x + 4| - 6 &=& 7 \end{eqnarray} $ Add ${6}$ to both sides: $ \begin{eqnarray} 5|x + 4| - 6 &=& 7 \\ \\ { + 6} &=& { + 6} \\ \\ 5|x + 4| &=& 13 \end{eqnarray} $ Divide both sides by ${5}$ $ \dfrac{5|x + 4|} {{5}} = \dfrac{13} {{5}} $ Simplify: $ |x + 4| = \dfrac{13}{5}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 4 = -\dfrac{13}{5} $ or $ x + 4 = \dfrac{13}{5} $ Solve for the solution where $x + 4$ is negative: $ x + 4 = -\dfrac{13}{5} $ Subtract ${4}$ from both sides: $ \begin{eqnarray} x + 4 &=& -\dfrac{13}{5} \\ \\ {- 4} && {- 4} \\ \\ x &=& -\dfrac{13}{5} - 4 \end{eqnarray} $ Change the ${ - 4}$ to an equivalent fraction with a denominator of $5$ $ x = - \dfrac{13}{5} {- \dfrac{20}{5}} $ $ x = -\dfrac{33}{5} $ Then calculate the solution where $x + 4$ is positive: $ x + 4 = \dfrac{13}{5} $ Subtract ${4}$ from both sides: $ \begin{eqnarray} x + 4 &=& \dfrac{13}{5} \\ \\ {- 4} && {- 4} \\ \\ x &=& \dfrac{13}{5} - 4 \end{eqnarray} $ Change the ${ - 4}$ to an equivalent fraction with a denominator of $5$ $ x = \dfrac{13}{5} {- \dfrac{20}{5}} $ $ x = -\dfrac{7}{5} $ Thus, the correct answer is $x = -\dfrac{33}{5} $ or $x = -\dfrac{7}{5} $.